In: Math
Aluminum, iron and magnesium are used to manufacture certain metal parts.The quantity of parts produced from x tons of aluminum y tons of iron and z tons of magnesium is Q (x, y, z) = xyz. Aluminum is $ 800 per ton, iron is $ 400 per ton, and magnesium is $ 600 per ton.
use the Lagrange multiplier method to determine the number of tons of each material that must be used to manufacture 5000 metal parts at the lowest possible cost.
Solution: Given the quantity Q of the metal parts produced from x, y, and z tons of aluminum, iron, and magnesium:
Since aluminum, iron, and magnesium are $800, $400, and $600 per ton, therefore, the cost function C will be given by:
... (1)
Now, the objective to determine the value of x, y, and z when quantity Q of the metal parts is equal to 5000 at the lowest possible cost by using the Lagrange multipliers. Therefore,
... (2)
Lagrange Multipliers: For the differentiable functions C(x,y,z) and Q(x,y,z), the extreme values of the function C subject to the condition Q can be obtained by solving the following equations simultaneously:
... (3)
... (4)
Also, the gradient of a function f is given by:
Therefore, the gradient of the function C and Q will be:
and
Substituting these values in the equation (3), we'll get:
Comparing each component, we'll get:
... (5)
... (6)
... (7)
Dividing equation (5) from (6) and (5) from (7), we'll get:
Substituting these values of y and z in the equation (2), since we've to calculate the minimum value of C w.r.t Q, we'll get
Therefore,
Therefore, we got the value of x, y, and z for the cost C to be minimum. Substituting these values in the equation (1), we'll get:
Therefore, 5000 parts can be manufactured at a minimum cost of $29592.
I hope it helps you!